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crystal structure and/or surface defects. London forces arise from
the spontaneous fluctuation of the electronic cloud in one atom
causing a corresponding fluctuation of electrons in its neighboring
atom, resulting in an attractive force between them. Importantly,
interparticle London forces are a long-range interaction, dependent
on the chemical composition and geometry of the interacting
nanocrystals and the interparticle distance between them. The
energy of London forces can be calculated according to Hamaker's
theory, which is based on the assumption that the dispersion potential
between two colloidal particles can be presented as a summation of
the dispersion interactions between pairs of atoms located within
the two particles.
31
The expression for London interaction energy
(E
) between two spheres is as follows:
L
È
RR
hRhRh
2
2
RR
hRhRhRR
hRh
2
˘
12
12
+
+
Í
Í
Í
Í
Í
˙
˙
˙
˙
˙
2
2
+ +
+ + +
2
2
2
4
A
1
2
1
2
12
H
E
=-
(13.3)
L
Ê
Ë
Á
2
ˆ
¯
˜
6
Rh
hRhRhRR
+
+ + +
+
2
2
1
2
ln
2
2
2
Î
˚
1
2
12
is the material-dependent Hamaker constant for the two
interacting spheres with radius R
where A
H
. h is the distance of the
closest approach between the two spheres.
and R
1
2
31
between two
nanocrystals is on the order of the thermal energy (3/2 k
The E
L
T). For
B
example, the E
between two 6 nm gold nanocrystals at 3.6 nm apart
L
∼
is
T at room temperature.
Electrostatic double-layer interactions are repulsive interactions
between charged nanocrystals, and the interaction energy can be
calculated on the basis of the theory of Debye and Hückel.
1.4 k
B
24
The
energy of electrostatic interactions between two identically charged
spheres is as follows:
È
-
k
˘
h
Rh
Rh
+
+
Re
y
2
=
peey
+
Uh
()
4
R
ln
1
(13.4)
Í
Í
˙
˙
el
0
s
+
2
Rh
Î
˚
e
e
where
is the dielectric constant of the solvent,
denotes the
0
vacuum permittivity,
stands for the surface electric potential, h is
the interparticle distance, and
y
s
k
is the inverse of the Debye screening
k
−
length:
1
= (1000
ee
k
T/N
e
2
I)
1/2
, where I is the ionic strength, k
0
B
A
B
is the Boltzmann constant, N
is the Avogadro's number, and e is the
A
24
According to this expression, a highly charged
surface and diluted electrolytes give long-range repulsion between
two colloidal nanocrystals.
elementary charge.
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